**Finding Inverse of a Function :**

Here we are going to see, how to find inverse of a function.

**Definition of f ^{−1 }:**

Suppose f is a one-to-one function.

To find a formula for f ^{−1}(y), solve the equation f(x) = y for x in terms of y.

In order to find the inverse of any function, we have to prove that the given function is one to one.

**Definition of one to one function :**

A function f is called one-to-one if for each number y in the range of f there is exactly one number x in the domain of f such that f(x) = y.

**Domain and range of inverse function :**

- The domain of f
^{−1}equals the range of f ; - the range of f
^{−1}equals the domain of f .

**Important note :**

The inverse function is not defined for a function that is not one-to-one.

Relationship between f and f^{−1}

Suppose f is a one-to-one function and x and y are numbers. Then

f(x) = y if and only if f^{ −1}(y) = x.

**Question 1 :**

Suppose f(x) = 2x + 3.

(a) Evaluate f^{−1}(11).

(b) Find a formula for f^{−1}(y).

**Solution :**

In order to find the inverse function, first let us prove the given function is one to one.

The domain of the function f(x) is all real values, each real values of x is associated with unique elements of y. So it is one to one function.

y = f(x) = 2x + 3

2x + 3 = y

2x = y - 3

x = (y - 3)/2

Replace x by f^{-1} (y) and y by x.

f^{-1}(y) = (y - 3)/2

**Question 2 :**

Suppose the domain of f is the interval [0, 2], with f defined on this domain by the equation f(x) = x^{2}.

(a) What is the range of f ?

(b) Find a formula for the inverse function f ^{−1}.

(c) What is the domain of the inverse function f^{ −1}?

(d) What is the range of the inverse function f^{ −1}?

**Solution :**

f(x) = x^{2}

If x = 0 f(0) = 0 |
If x = 1 f(1) = 1 |
If x = 2 f(2) = 4 |

From this, we know that range is [0, 4].

(b) Find a formula for the inverse function f ^{−1}.

From the table given above, we know that each values is associated with unique elements. So it is one to one.

y = x^{2}

x = √y

Here x can be replaced by f^{-1}(y), so f^{-1}(y) = √y

(c) What is the domain of the inverse function f^{ −1}?

Domain of f^{ −1} = Range of f = [0, 4]

(d) What is the range of the inverse function f^{ −1}?

Range of f^{ −1} = Domain of f = [0, 2]

After having gone through the stuff given above, we hope that the students would have understood "Finding Inverse of a Function".

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